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Implementing real polyhedral homotopy

Kisun Lee, Julia Lindberg, Jose Israel Rodriguez

TL;DR

This work targets the efficient computation of all real solutions to sparse polynomial systems by implementing a real polyhedral homotopy in Julia, grounded in ergur2019polyhedral. It delivers an end-to-end workflow based on three core functions—certify_patchwork, generate_binomials, and rph_track—enabling certificate-based, real-solution-focused homotopy tracking. The implementation leverages tropical-geometric concepts such as regular subdivisions, Cayley configurations, and mixed cells, and relies on numerical-certification techniques to provide guarantees where possible. The resulting RealPolyhedralHomotopy.jl package provides the first homotopy-based tool that provably finds all real solutions for patchworked systems and serves as a practical heuristic for non-patchworked cases, with potential impact in fields ranging from economics to reaction networks and optimization.

Abstract

We implement a real polyhedral homotopy method using three functions. The first function provides a certificate that our real polyhedral homotopy is applicable to a given system; the second function generates binomial systems for a start system; the third function outputs target solutions from the start system obtained by the second function. This work realizes the theoretical contributions in \cite{ergur2019polyhedral} as easy to use functions, allowing for further investigation into real homotopy algorithms.

Implementing real polyhedral homotopy

TL;DR

This work targets the efficient computation of all real solutions to sparse polynomial systems by implementing a real polyhedral homotopy in Julia, grounded in ergur2019polyhedral. It delivers an end-to-end workflow based on three core functions—certify_patchwork, generate_binomials, and rph_track—enabling certificate-based, real-solution-focused homotopy tracking. The implementation leverages tropical-geometric concepts such as regular subdivisions, Cayley configurations, and mixed cells, and relies on numerical-certification techniques to provide guarantees where possible. The resulting RealPolyhedralHomotopy.jl package provides the first homotopy-based tool that provably finds all real solutions for patchworked systems and serves as a practical heuristic for non-patchworked cases, with potential impact in fields ranging from economics to reaction networks and optimization.

Abstract

We implement a real polyhedral homotopy method using three functions. The first function provides a certificate that our real polyhedral homotopy is applicable to a given system; the second function generates binomial systems for a start system; the third function outputs target solutions from the start system obtained by the second function. This work realizes the theoretical contributions in \cite{ergur2019polyhedral} as easy to use functions, allowing for further investigation into real homotopy algorithms.
Paper Structure (16 sections, 3 theorems, 23 equations, 1 figure)

This paper contains 16 sections, 3 theorems, 23 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Regular subdivisions and triangulations
  4. Cayley configurations and mixed cells
  5. --discriminant and its amoeba
  6. Homotopy continuation
  7. Polyhedral homotopy continuation and Bernstein's theorem
  8. Certification of numerical solutions
  9. Finding a real polyhedral homotopy
  10. Real polyhedral homotopy implementation
  11. certify_patchwork
  12. generate_binomials
  13. rph_track
  14. An rph_track option
  15. Reliability
  16. ...and 1 more sections

Key Result

Theorem 2.3

bernshtein1975number Let $F$ be a system of polynomials $f_1,\dots, f_n$ in $\mathbb{C}[x_1,\dots, x_n]$. For Newton polytopes $Q_{f_i}$ for each $f_i$, we have Furthermore, for polynomials $f_1,\dots, f_n$ with generic coefficients the inequality is tight.

Figures (1)

  • Figure 1: The polytope $A^w$ from \ref{['ex:def1']}.

Theorems & Definitions (11)

  • Example 2.1
  • Example 2.2
  • Theorem 2.3: Bernstein's theorem
  • Definition 2.4: Polyhedral homotopy
  • Proposition 3.1
  • Proposition 3.2
  • Definition 3.3: Real polyhedral homotopy
  • Example 3.4
  • Remark 4.1
  • Remark 4.2
  • ...and 1 more